# Ask for a counter-example or ideas for a test

Leave $$A$$ be an upper triangular matrix of blocks; The blocks in the diagonal being. $$a in M_p ({ mathbb C})$$ Y $$c in M_q ({ mathbb C})$$.

Leave $$P$$ be a polynomial such that $$P (A) = 0_n$$.
Leave $$(i, j)$$ be two such indices $$j le p .
It seems that $$dP (A) cdot E_ {ij} = 0$$.

Thanks for your interest.